Restriction of Automorphisms of the Unit Disc

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1. If $F$ is a finite subset of the unit disc $B(0,1)$ and $U$= $B(0,1)$ $F$,
   then show that every analytic automorphism of $U$ is the restriction to $U$ of an analytic automorphism of $B(0,1)$ that permutes the points of $F$.




2. If r $in mathbbR$ with $0<|r|<1$, show that $exists$ at most one real $s in mathbbR$ with $0<s<1$ for which $V_s$ = $B(0,1)$ $0,r,s$ admits of an analytic automorphism other than the identity.

The first part of the problem has been proved. For the second part, firstly I have some confusion regarding what is needed to prove. I think we need to prove that for given $r in mathbbR$, if $exists$ distinct $0<s$, $t<1$ such that both $V_s$ and $V_t$ admit an analytic automorphism, then one of them must be identity. If this is what is $only$ required to be proved, I was trying to use the form of the analytic automorphisms of the disc and the first part of the problem to prove this result, but to no avail. Thanks for any help.

complex-analysis

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edited Jul 16 at 7:58

asked Jul 15 at 16:04

Ester

8791925

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Let $f: U to U$ be an automorphism. Writing $f: U to U hookrightarrow B(0,1)$, we see that the $f$ is bounded near its singularities. Thus, by Riemann's theorem, it extends to a map $tildef: B(0,1) to B(0,1)$. This $tildef$ must be an automorphism (since it must map $F$ into $F$).
  – Sameer Kailasa
  Jul 15 at 17:31
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks.Any ideas about the second part.
  – Ester
  Jul 15 at 18:03
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  For the second part, we can use the classification of automorphisms of the unit disk: any automorphism $f: B(0,1) to B(0,1)$ is of the form $$f(z) = e^itheta fracz-aoverlinea z - 1$$ for some $ain mathbbC$ with $|a|<1$. If $f$ is then an automorphism of $V$, then the very specific function above has to permute $0,r,s$, which is quite a restriction.
  – Sameer Kailasa
  Jul 15 at 18:11
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Can you please give a complete solution so that I can accept.That will be very helpful.
  – Ester
  Jul 15 at 18:12
  

  
  
  
  

add a comment | 

up vote
0
down vote

favorite

1. If $F$ is a finite subset of the unit disc $B(0,1)$ and $U$= $B(0,1)$ $F$,
   then show that every analytic automorphism of $U$ is the restriction to $U$ of an analytic automorphism of $B(0,1)$ that permutes the points of $F$.




2. If r $in mathbbR$ with $0<|r|<1$, show that $exists$ at most one real $s in mathbbR$ with $0<s<1$ for which $V_s$ = $B(0,1)$ $0,r,s$ admits of an analytic automorphism other than the identity.

The first part of the problem has been proved. For the second part, firstly I have some confusion regarding what is needed to prove. I think we need to prove that for given $r in mathbbR$, if $exists$ distinct $0<s$, $t<1$ such that both $V_s$ and $V_t$ admit an analytic automorphism, then one of them must be identity. If this is what is $only$ required to be proved, I was trying to use the form of the analytic automorphisms of the disc and the first part of the problem to prove this result, but to no avail. Thanks for any help.

complex-analysis

share|cite|improve this question

edited Jul 16 at 7:58

asked Jul 15 at 16:04

Ester

8791925

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Let $f: U to U$ be an automorphism. Writing $f: U to U hookrightarrow B(0,1)$, we see that the $f$ is bounded near its singularities. Thus, by Riemann's theorem, it extends to a map $tildef: B(0,1) to B(0,1)$. This $tildef$ must be an automorphism (since it must map $F$ into $F$).
  – Sameer Kailasa
  Jul 15 at 17:31
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks.Any ideas about the second part.
  – Ester
  Jul 15 at 18:03
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  For the second part, we can use the classification of automorphisms of the unit disk: any automorphism $f: B(0,1) to B(0,1)$ is of the form $$f(z) = e^itheta fracz-aoverlinea z - 1$$ for some $ain mathbbC$ with $|a|<1$. If $f$ is then an automorphism of $V$, then the very specific function above has to permute $0,r,s$, which is quite a restriction.
  – Sameer Kailasa
  Jul 15 at 18:11
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Can you please give a complete solution so that I can accept.That will be very helpful.
  – Ester
  Jul 15 at 18:12
  

  
  
  
  

add a comment | 

up vote
0
down vote

favorite

up vote
0
down vote

favorite

1. If $F$ is a finite subset of the unit disc $B(0,1)$ and $U$= $B(0,1)$ $F$,
   then show that every analytic automorphism of $U$ is the restriction to $U$ of an analytic automorphism of $B(0,1)$ that permutes the points of $F$.




2. If r $in mathbbR$ with $0<|r|<1$, show that $exists$ at most one real $s in mathbbR$ with $0<s<1$ for which $V_s$ = $B(0,1)$ $0,r,s$ admits of an analytic automorphism other than the identity.

The first part of the problem has been proved. For the second part, firstly I have some confusion regarding what is needed to prove. I think we need to prove that for given $r in mathbbR$, if $exists$ distinct $0<s$, $t<1$ such that both $V_s$ and $V_t$ admit an analytic automorphism, then one of them must be identity. If this is what is $only$ required to be proved, I was trying to use the form of the analytic automorphisms of the disc and the first part of the problem to prove this result, but to no avail. Thanks for any help.

complex-analysis

share|cite|improve this question

edited Jul 16 at 7:58

asked Jul 15 at 16:04

Ester

8791925

1. If $F$ is a finite subset of the unit disc $B(0,1)$ and $U$= $B(0,1)$ $F$,
   then show that every analytic automorphism of $U$ is the restriction to $U$ of an analytic automorphism of $B(0,1)$ that permutes the points of $F$.




2. If r $in mathbbR$ with $0<|r|<1$, show that $exists$ at most one real $s in mathbbR$ with $0<s<1$ for which $V_s$ = $B(0,1)$ $0,r,s$ admits of an analytic automorphism other than the identity.

The first part of the problem has been proved. For the second part, firstly I have some confusion regarding what is needed to prove. I think we need to prove that for given $r in mathbbR$, if $exists$ distinct $0<s$, $t<1$ such that both $V_s$ and $V_t$ admit an analytic automorphism, then one of them must be identity. If this is what is $only$ required to be proved, I was trying to use the form of the analytic automorphisms of the disc and the first part of the problem to prove this result, but to no avail. Thanks for any help.

complex-analysis

share|cite|improve this question

edited Jul 16 at 7:58

asked Jul 15 at 16:04

Ester

8791925

share|cite|improve this question

share|cite|improve this question

edited Jul 16 at 7:58

edited Jul 16 at 7:58

edited Jul 16 at 7:58

asked Jul 15 at 16:04

Ester

8791925

asked Jul 15 at 16:04

Ester

8791925

asked Jul 15 at 16:04

Ester

8791925

8791925

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Let $f: U to U$ be an automorphism. Writing $f: U to U hookrightarrow B(0,1)$, we see that the $f$ is bounded near its singularities. Thus, by Riemann's theorem, it extends to a map $tildef: B(0,1) to B(0,1)$. This $tildef$ must be an automorphism (since it must map $F$ into $F$).
  – Sameer Kailasa
  Jul 15 at 17:31
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks.Any ideas about the second part.
  – Ester
  Jul 15 at 18:03
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  For the second part, we can use the classification of automorphisms of the unit disk: any automorphism $f: B(0,1) to B(0,1)$ is of the form $$f(z) = e^itheta fracz-aoverlinea z - 1$$ for some $ain mathbbC$ with $|a|<1$. If $f$ is then an automorphism of $V$, then the very specific function above has to permute $0,r,s$, which is quite a restriction.
  – Sameer Kailasa
  Jul 15 at 18:11
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Can you please give a complete solution so that I can accept.That will be very helpful.
  – Ester
  Jul 15 at 18:12
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Let $f: U to U$ be an automorphism. Writing $f: U to U hookrightarrow B(0,1)$, we see that the $f$ is bounded near its singularities. Thus, by Riemann's theorem, it extends to a map $tildef: B(0,1) to B(0,1)$. This $tildef$ must be an automorphism (since it must map $F$ into $F$).
  – Sameer Kailasa
  Jul 15 at 17:31
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks.Any ideas about the second part.
  – Ester
  Jul 15 at 18:03
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  For the second part, we can use the classification of automorphisms of the unit disk: any automorphism $f: B(0,1) to B(0,1)$ is of the form $$f(z) = e^itheta fracz-aoverlinea z - 1$$ for some $ain mathbbC$ with $|a|<1$. If $f$ is then an automorphism of $V$, then the very specific function above has to permute $0,r,s$, which is quite a restriction.
  – Sameer Kailasa
  Jul 15 at 18:11
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Can you please give a complete solution so that I can accept.That will be very helpful.
  – Ester
  Jul 15 at 18:12
  

  
  
  
  

1

1

Let $f: U to U$ be an automorphism. Writing $f: U to U hookrightarrow B(0,1)$, we see that the $f$ is bounded near its singularities. Thus, by Riemann's theorem, it extends to a map $tildef: B(0,1) to B(0,1)$. This $tildef$ must be an automorphism (since it must map $F$ into $F$).
– Sameer Kailasa
Jul 15 at 17:31

Let $f: U to U$ be an automorphism. Writing $f: U to U hookrightarrow B(0,1)$, we see that the $f$ is bounded near its singularities. Thus, by Riemann's theorem, it extends to a map $tildef: B(0,1) to B(0,1)$. This $tildef$ must be an automorphism (since it must map $F$ into $F$).
– Sameer Kailasa
Jul 15 at 17:31

Thanks.Any ideas about the second part.
– Ester
Jul 15 at 18:03

Thanks.Any ideas about the second part.
– Ester
Jul 15 at 18:03

For the second part, we can use the classification of automorphisms of the unit disk: any automorphism $f: B(0,1) to B(0,1)$ is of the form $$f(z) = e^itheta fracz-aoverlinea z - 1$$ for some $ain mathbbC$ with $|a|<1$. If $f$ is then an automorphism of $V$, then the very specific function above has to permute $0,r,s$, which is quite a restriction.
– Sameer Kailasa
Jul 15 at 18:11

For the second part, we can use the classification of automorphisms of the unit disk: any automorphism $f: B(0,1) to B(0,1)$ is of the form $$f(z) = e^itheta fracz-aoverlinea z - 1$$ for some $ain mathbbC$ with $|a|<1$. If $f$ is then an automorphism of $V$, then the very specific function above has to permute $0,r,s$, which is quite a restriction.
– Sameer Kailasa
Jul 15 at 18:11

Can you please give a complete solution so that I can accept.That will be very helpful.
– Ester
Jul 15 at 18:12

Can you please give a complete solution so that I can accept.That will be very helpful.
– Ester
Jul 15 at 18:12

add a comment | 

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